Find the length of the line PA.

RO divides the rectangle into two congruent right triangles.
The area of the one triangle is equal half area of the rectangle.
Calculate the area of rectangle:
[tex]A_R=lw\\l=12,\ w=5\\\\A_R=12\cdot5=60[/tex]
The area of right triangle:
[tex]A_T=\dfrac{1}{2}A_R\to A_T=\dfrac{1}{2}\cdot60=30[/tex]
Use the Pythagorean theorem to calculate the length of RO:
[tex]|RO|^2=5^2+12^2\\\\|RO|^2=25+144\\\\|RO|^2=169\to|RO|=\sqrt{169}\to|RO|=13[/tex]
The formula of an area of this right triangle is:
[tex]A_T=\dfrac{1}{2}|RO||PA|[/tex]
Therefore we have the equation:
[tex]\dfrac{1}{2}(13)|PA|=30\qquad|\cdot2\\\\13|PA|=60\quad|:13\\\\|PA|=\dfrac{60}{13}\\\\|PA|\approx4.62[/tex]